Finite Group Databasebeta

← #120431

Subgroups — #120431

Order 512.

458 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21989C211yesmaximal, characteristic
2#17822C211yesmaximal, characteristic
3#10848C211yesmaximal, characteristic
4#7993C211yesmaximal, characteristic
5#7974C211yesmaximal, characteristic
6#7960C211yesmaximal, characteristic
7#7953C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1949—21no—
16#1949—21no—
17#1854—21no—
18#1854—21no—
19#1847—21no—
20#1836—21no—
21C2×#344—21no—
22C2×#344—21no—
23#1573—22no—
24#1571—21no—
25#1570—21no—
26#1569C414yescomplemented
27#1291—22no—
28#1288—22no—
29#1286—21no—
30#1286—21no—
31#1285—22no—
32C22×M16—21no—
33C22×M16—21no—
34C2×#117D411yescharacteristic
35C2×#117D411yescharacteristic
36C22×C42C2311yescharacteristic
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432C2×C412yes—
42#432D412yes—
43#432D412yes—
44#432Q812yes—
45#431—21no—
46#431—21no—
47#431—21no—
48#431Q812yes—
49C2×#107—21no—
50C2×#107—21no—
51C2×#106—21no—
52C2×#106—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423C2×C412yes—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8D411yescharacteristic
62C2×C4⋊C8D411yescharacteristic
63#407—21no—
64#407—21no—
65C2×C22⋊C8—21no—
66C4×M16—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16C2×C412yes—
72C2×#97—21no—
73C2×#97—21no—
74C2×C4×C8Q812yes—
75#384—21no—
76#383—21no—
77#382—21no—
78#382D411yescharacteristic
79#382D411yescharacteristic
80#381—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C22⋊C4—21no—
88#376—21no—
89#344—42no—
90#344—42no—
91#340—21no—
92#340—21no—
93#340—21no—
94#340—21no—
95#335—24no—
96#334—24no—
97C23×C4C2×D411yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—41no—
118#117—41no—
119#117—41no—
120#117—41no—
121#117—41no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—41no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—41no—
134C2×C42C22×C411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×Q811yescharacteristic
140C2×C42C2×Q811yescharacteristic
141C2×C42C2×Q811yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146#107—81no—
147#107—81no—
148#106—81no—
149#106—81no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8C22⋊C412yes—
161C4⋊C8C22⋊C412yes—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—41no—
168#97—22no—
169#97—22no—
170#97—41no—
171#97—41no—
172#97—41no—
173#97C4⋊C414yes—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186C24#12311yescharacteristic
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4#11711yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4C4×Q811yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C22⋊Q811yescharacteristic
206C22×C4C22⋊Q811yescharacteristic
207C22×C4C22⋊Q811yescharacteristic
208C22×C4#12311yescharacteristic
209C22×C4#12411yescharacteristic
210C22×C4#12511yescharacteristic
211C22×C4#12511yescharacteristic
212C22×C4#12511yescharacteristic
213C22×C4#12611yescharacteristic
214C22×C4C4⋊Q811yescharacteristic
215M16—42no—
216M16—42no—
217M16—42no—
218M16—42no—
219M16—42no—
220M16—42no—
221M16—82no—
222M16—82no—
223M16—81no—
224M16—81no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—81no—
238C2×C8—81no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—82no—
252C42—81no—
253C42—81no—
254C42—82no—
255C42C2×C22⋊C411yescharacteristic
256C42C2×C4⋊C411yescharacteristic
257C42C4×D411yescharacteristic
258C42C4×Q811yescharacteristic
259C42#12111yescharacteristic
260C42#12111yescharacteristic
261C42C22⋊Q811yescharacteristic
262C42C22⋊Q811yescharacteristic
263C42C22⋊Q811yescharacteristic
264C42C22⋊Q811yescharacteristic
265C42C22⋊Q811yescharacteristic
266C42#12311yescharacteristic
267C42#12511yescharacteristic
268C42#12611yescharacteristic
269C42C4⋊Q811yescharacteristic
270C42C4⋊Q811yescharacteristic
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#38711yescharacteristic
276C23#39411yescharacteristic
277C23#39811yescharacteristic
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4#38011yescharacteristic
313C2×C4#38611yescharacteristic
314C2×C4#38711yescharacteristic
315C2×C4#38911yescharacteristic
316C2×C4#38911yescharacteristic
317C2×C4#39111yescharacteristic
318C2×C4#39511yescharacteristic
319C2×C4#39611yescharacteristic
320C2×C4#39611yescharacteristic
321C2×C4#39811yescharacteristic
322C2×C4#39811yescharacteristic
323C2×C4#40011yescharacteristic
324C2×C4#45311yescharacteristic
325C2×C4#45411yescharacteristic
326C2×C4#45511yescharacteristic
327C2×C4#45611yescharacteristic
328C8—44no—
329C8—44no—
330C8—42no—
331C8—42no—
332C8—82no—
333C8—82no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—81no—
343C22—81no—
344C22#144111yescharacteristic
345C22#203711yescharacteristic
346C22#203811yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—161no—
358C4—161no—
359C4—161no—
360C4—161no—
361C4#188211yescharacteristic
362C4#188311yescharacteristic
363C4#206011yescharacteristic
364C4#206111yescharacteristic
365C2—21no—
366C2—21no—
367C2—41no—
368C2—41no—
369C2#1121711yescharacteristic
370C2#1124511yescharacteristic
371C2#1125111yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
26. #1569C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#219892
#178222
#108482
#79932
#79742
#79602
#79532